Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 36\cdot 37 + 26\cdot 37^{2} + 31\cdot 37^{3} + 35\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 32\cdot 37 + 3\cdot 37^{2} + 6\cdot 37^{3} + 32\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 8 + 37 + 14\cdot 37^{2} + 6\cdot 37^{3} + 24\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 + 5\cdot 37 + 4\cdot 37^{2} + 19\cdot 37^{3} + 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 + 31\cdot 37 + 32\cdot 37^{2} + 17\cdot 37^{3} + 35\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 + 35\cdot 37 + 22\cdot 37^{2} + 30\cdot 37^{3} + 12\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 31 + 4\cdot 37 + 33\cdot 37^{2} + 30\cdot 37^{3} + 4\cdot 37^{4} +O\left(37^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 35 + 10\cdot 37^{2} + 5\cdot 37^{3} + 37^{4} +O\left(37^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3,8,6)(2,5,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
